Few-electron atomic ions in non-relativistic QED: The ground state
Following detailed analysis of relativistic, QED and mass corrections for helium-like and lithium-like ions with static nuclei for Z≤20 the domain of applicability of Non-Relativistic QED (NRQED) is localized for ground state energy. It is demonstrated that for both helium-like and lithium-like ions...
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| Published in: | Annals of physics Vol. 409; p. 167908 |
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| Abstract | Following detailed analysis of relativistic, QED and mass corrections for helium-like and lithium-like ions with static nuclei for Z≤20 the domain of applicability of Non-Relativistic QED (NRQED) is localized for ground state energy. It is demonstrated that for both helium-like and lithium-like ions with Z≤20 the finite nuclear mass effects do not change 4–5 significant digits (s.d.), and the leading relativistic and QED effects leave unchanged 3–4 s.d. in the ground state energy. It is shown that the non-relativistic ground state energy can be interpolated with accuracy not less than 13 s.d. for Z≤12, and not less than 12 s.d. for Z≤50 for helium-like as well as for Z≤20 for lithium-like ions by a compact meromorphic function in variable λ=Z−ZB (here ZB is the 2nd critical charge (Turbiner et al., 2016)), P9(λ)∕Q5(λ). It is found that both the Majorana formula – a second degree polynomial in Z with two free parameters – and a fourth degree polynomial in λ (a generalization of the Majorana formula) reproduce the ground state energy of the helium-like and lithium-like ions for Z≤20 in the domain of applicability of NRQED, thus, at least, 3 s.d. It is noted that ≳99.9% of the ground state energy is given by the variational energy for properly optimized trial function of the form of (anti)-symmetrized product of three (six) screened Coulomb orbitals for two-(three) electron system with 3 (7) free parameters for Z≤20, respectively. It may imply that these trial functions are, in fact, exact wavefunctions in non-relativistic QED, thus, the NRQED effective potential can be derived. It is shown that the sum of relativistic and QED effects in leading approximation - 3 s.d. - for both 2 and 3 electron systems is interpolated by 4th degree polynomial in Z for Z≤20. |
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| AbstractList | Following detailed analysis of relativistic, QED and mass corrections for helium-like and lithium-like ions with static nuclei for Z≤20 the domain of applicability of Non-Relativistic QED (NRQED) is localized for ground state energy. It is demonstrated that for both helium-like and lithium-like ions with Z≤20 the finite nuclear mass effects do not change 4–5 significant digits (s.d.), and the leading relativistic and QED effects leave unchanged 3–4 s.d. in the ground state energy. It is shown that the non-relativistic ground state energy can be interpolated with accuracy not less than 13 s.d. for Z≤12, and not less than 12 s.d. for Z≤50 for helium-like as well as for Z≤20 for lithium-like ions by a compact meromorphic function in variable λ=Z−ZB (here ZB is the 2nd critical charge (Turbiner et al., 2016)), P9(λ)∕Q5(λ). It is found that both the Majorana formula – a second degree polynomial in Z with two free parameters – and a fourth degree polynomial in λ (a generalization of the Majorana formula) reproduce the ground state energy of the helium-like and lithium-like ions for Z≤20 in the domain of applicability of NRQED, thus, at least, 3 s.d. It is noted that ≳99.9% of the ground state energy is given by the variational energy for properly optimized trial function of the form of (anti)-symmetrized product of three (six) screened Coulomb orbitals for two-(three) electron system with 3 (7) free parameters for Z≤20, respectively. It may imply that these trial functions are, in fact, exact wavefunctions in non-relativistic QED, thus, the NRQED effective potential can be derived. It is shown that the sum of relativistic and QED effects in leading approximation - 3 s.d. - for both 2 and 3 electron systems is interpolated by 4th degree polynomial in Z for Z≤20. Following detailed analysis of relativistic, QED and mass corrections for helium-like and lithium-like ions with static nuclei for Z≤20 the domain of applicability of Non-Relativistic QED (NRQED) is localized for ground state energy. It is demonstrated that for both helium-like and lithium-like ions with Z≤20 the finite nuclear mass effects do not change 4–5 significant digits (s.d.), and the leading relativistic and QED effects leave unchanged 3–4 s.d. in the ground state energy. It is shown that the non-relativistic ground state energy can be interpolated with accuracy not less than 13 s.d. for Z≤12, and not less than 12 s.d. for Z≤50 for helium-like as well as for Z≤20 for lithium-like ions by a compact meromorphic function in variable λ=√(Z−Z{sub B}) (here Z{sub B} is the 2nd critical charge (Turbiner et al., 2016)), P{sub 9}(λ)∕Q{sub 5}(λ). It is found that both the Majorana formula – a second degree polynomial in Z with two free parameters – and a fourth degree polynomial in λ (a generalization of the Majorana formula) reproduce the ground state energy of the helium-like and lithium-like ions for Z≤20 in the domain of applicability of NRQED, thus, at least, 3 s.d. It is noted that ≳99.9% of the ground state energy is given by the variational energy for properly optimized trial function of the form of (anti)-symmetrized product of three (six) screened Coulomb orbitals for two-(three) electron system with 3 (7) free parameters for Z≤20, respectively. It may imply that these trial functions are, in fact, exact wavefunctions in non-relativistic QED, thus, the NRQED effective potential can be derived. It is shown that the sum of relativistic and QED effects in leading approximation - 3 s.d. - for both 2 and 3 electron systems is interpolated by 4th degree polynomial in Z for Z≤20. |
| ArticleNumber | 167908 |
| Author | Vieyra, Juan Carlos Lopez Turbiner, Alexander V. Olivares-Pilón, Horacio |
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| Cites_doi | 10.1007/s10701-012-9685-1 10.3367/UFNr.0144.198409b.0035 10.1063/1.1727380 10.1063/1.2801981 10.1016/j.physrep.2014.11.006 10.1103/PhysRevA.57.1652 10.1088/0953-4075/34/6/308 10.1103/PhysRevA.61.064503 10.1142/S021773231650156X 10.1103/PhysRevLett.112.173001 10.1002/qua.22217 10.1088/1751-8121/aa6cc2 10.1103/PhysRevA.78.052511 10.1016/j.nuclphysa.2003.11.003 10.1103/PhysRevA.10.1109 10.1103/PhysRevA.55.1820 10.1007/BF01375457 10.1016/S0370-1573(00)00077-6 10.1139/p88-100 10.1103/PhysRevA.95.062510 10.1139/cjp-2015-0366 10.1016/j.physleta.2014.12.029 10.1016/0022-2852(62)90021-8 10.1038/nature13026 10.1016/j.aop.2018.04.021 10.1088/0953-4075/36/1/311 10.1103/PhysRevA.36.1013 |
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| SubjectTerms | ACCURACY ATOMIC IONS CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS ELECTRONS GROUND STATES HELIUM LITHIUM MAJORANA SPINORS NUCLEI POLYNOMIALS QUANTUM ELECTRODYNAMICS RELATIVISTIC RANGE VARIATIONAL METHODS WAVE FUNCTIONS |
| Title | Few-electron atomic ions in non-relativistic QED: The ground state |
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